FLINT, Mich.—Like most people who enter the Mission of Hope shelter these days, Catherine Brown comes for the water.

Ms. Brown, 55 years old, was once an aide to former Mayor Don Williamson but has been unemployed since she was laid off in 2008. Now, she is one of the 100,000 Flint residents whose lives have been upended by lead contamination of the city’s water. She washes herself with bottled water to avoid rashes she got from the…